3.356 \(\int \sec ^3(x) \tan (x) \, dx\)

Optimal. Leaf size=8 \[ \frac {\sec ^3(x)}{3} \]

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Rubi [A]  time = 0.01, antiderivative size = 8, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2606, 30} \[ \frac {\sec ^3(x)}{3} \]

Antiderivative was successfully verified.

[In]

Int[Sec[x]^3*Tan[x],x]

[Out]

Sec[x]^3/3

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2606

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[a/f, Subst[
Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n -
1)/2] &&  !(IntegerQ[m/2] && LtQ[0, m, n + 1])

Rubi steps

\begin {align*} \int \sec ^3(x) \tan (x) \, dx &=\operatorname {Subst}\left (\int x^2 \, dx,x,\sec (x)\right )\\ &=\frac {\sec ^3(x)}{3}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 8, normalized size = 1.00 \[ \frac {\sec ^3(x)}{3} \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[x]^3*Tan[x],x]

[Out]

Sec[x]^3/3

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sec ^3(x) \tan (x) \, dx \]

Verification is Not applicable to the result.

[In]

IntegrateAlgebraic[Sec[x]^3*Tan[x],x]

[Out]

Could not integrate

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fricas [A]  time = 0.62, size = 6, normalized size = 0.75 \[ \frac {1}{3 \, \cos \relax (x)^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)^3*tan(x),x, algorithm="fricas")

[Out]

1/3/cos(x)^3

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giac [A]  time = 0.63, size = 6, normalized size = 0.75 \[ \frac {1}{3 \, \cos \relax (x)^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)^3*tan(x),x, algorithm="giac")

[Out]

1/3/cos(x)^3

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maple [A]  time = 0.06, size = 7, normalized size = 0.88

method	result	size
derivativedivides	\(\frac {\left (\sec ^{3}\relax (x )\right )}{3}\)	\(7\)
default	\(\frac {\left (\sec ^{3}\relax (x )\right )}{3}\)	\(7\)
risch	\(\frac {8 \,{\mathrm e}^{3 i x}}{3 \left (1+{\mathrm e}^{2 i x}\right )^{3}}\)	\(17\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(x)^3*tan(x),x,method=_RETURNVERBOSE)

[Out]

1/3*sec(x)^3

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maxima [A]  time = 0.47, size = 6, normalized size = 0.75 \[ \frac {1}{3 \, \cos \relax (x)^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)^3*tan(x),x, algorithm="maxima")

[Out]

1/3/cos(x)^3

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mupad [B]  time = 0.32, size = 6, normalized size = 0.75 \[ \frac {1}{3\,{\cos \relax (x)}^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(x)/cos(x)^3,x)

[Out]

1/(3*cos(x)^3)

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sympy [A]  time = 0.07, size = 7, normalized size = 0.88 \[ \frac {1}{3 \cos ^{3}{\relax (x )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)**3*tan(x),x)

[Out]

1/(3*cos(x)**3)

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